PHYS 1441 – Section 001
Lecture #5
Tuesday, June 10, 2014
Dr. Jaehoon Yu
•
•
•
•
•
•
Trigonometry Refresher
Properties and operations of vectors
Components of the 2D Vector
Understanding the 2 Dimensional Motion
2D Kinematic Equations of Motion
Projectile Motion
Today’s homework is homework #3, due 11pm, Friday, June 13!!
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
1
Announcements
• Reading Assignment
– CH3.7
• Quiz #2
– Beginning of the class this Thursday, June, 12
– Covers CH3.1 to what we finish tomorrow, Wednesday,
June 11
– Bring your calculator but DO NOT input formula into it!
– You can prepare a one 8.5x11.5 sheet (front and back) of
handwritten formulae and values of constants for the
exam  no solutions, derivations or definitions!
• No additional formulae or values of constants will be provided!
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
2
Special Project #2 for Extra Credit
• Show that the trajectory of a projectile motion
is a parabola!!
– 20 points
– Due: Monday, June 16
– You MUST show full details of your OWN
computations to obtain any credit
• Beyond what was covered in this lecture note
and in the book!
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
3
• Definitions of sinθ, cos and tan
sin  
cos 
Length of the opposite side to q
h
= o
Length of the hypotenuse of the right triangle
h
Length of the adjacent side to 

Length of the hypotenuse of the right triangle
ha

h
tan  
Solve for θ
-1 æ ha ö
cos
q=
çè h ÷ø

ha=length of the side adjacent to the angle 
ho
Length of the opposite side to 

Length of the adjacent side to 
ha
sin 

tan  
cos 
Tuesday, June 10, 2014
ho
h
ha
h
ho

ha
90o
ho=length of the side
opposite to the angle
Trigonometry Refresher
Solve for θ
æ ho ö
q = tan ç ÷
è ha ø
-1
Pythagorian theorem: For right angle triangles
h 2  ho2  ha2
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
h  ho2  ha2
4
Vector and Scalar
Vector quantities have both magnitudes (sizes)
and directions Velocity, acceleration, force, momentum, etc
Normally denoted in BOLD letters, F, or a letter with arrow on top
Their sizes or magnitudes are denoted with normal letters, F, or
absolute values:
Scalar quantities have magnitudes only
Can be completely specified with a value
and its unit Normally denoted in normal letters, E
Speed, energy,
heat, mass,
time, etc
Both have units!!!
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
5
Properties of Vectors
• Two vectors are the same if their sizes and the directions
are the same, no matter where they are on a coordinate
system!!  You can move them around as you wish as
long as their directions and sizes are kept the same.
Which ones are the
same vectors?
y
D
A=B=E=D
F
A
Why aren’t the others?
B
x
E
Tuesday, June 10, 2014
C
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
C: The same magnitude
but opposite direction:
C=-A:A negative vector
F: The same direction
but different magnitude
6
•
Vector Operations
Addition:
– Triangular Method: One can add vectors by connecting the head of one vector to
the tail of the other (head-to-tail)
– Parallelogram method: Connect the tails of the two vectors and extend
– Addition is commutative: Changing order of operation does not affect the results
A+B=B+A, A+B+C+D+E=E+C+A+B+D
A+B
B
A
•
A
=
B
A+B
OR
A+B
B
A
Subtraction:
– The same as adding a negative vector:A - B = A + (-B)
A
A-B
•
-B
Since subtraction is the equivalent to adding a
negative vector, subtraction is also commutative!!!
Multiplication by a scalar is
increasing the magnitude A, B=2A
Tuesday, June 10, 2014
A
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
B=2A
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Example for Vector Addition
A car travels 20.0km due north followed by 35.0km in a direction 60.0o West
of North. Find the magnitude and direction of resultant displacement.
Bsin60o N
r
2
2
= A2 + B2 ( cos2 q + sin 2 q ) + 2AB cosq
B 60o Bcos60o

( A + Bcosq ) + ( Bsinq )
r=
= A2 + B2 + 2ABcosq
=
20
A
( 20.0 )2 + ( 35.0 )2 + 2 ´ 20.0 ´ 35.0 cos60
= 2325 = 48.2(km)
E
q = tan
B sin 60
-1
A + B cos 60
Do this using
35.0 sin 60
= tan
components!!
20.0 + 35.0 cos 60
30.3
= tan -1
= 38.9 to W wrt N
37.5
-1
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
8
Components and Unit Vectors
Coordinate systems are useful in expressing vectors in their components
y
Ay
(-,+)
(+,+)
A
(Ax,Ay)
Tuesday, June 10, 2014
(+,-)
}
Ay 

(-,-)
Ax 
Ax
x
A  Ax  Ay
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
2
2
Components
} Magnitude
9
Unit Vectors
• Unit vectors are the ones that tells us the
directions of the components
– Very powerful and makes vector notation and operations
much easier!
• Dimensionless
• Magnitudes these vectors are exactly 1
• Unit vectors are usually expressed in i, j, k or
So a vector A can be
expressed as
Tuesday, June 10, 2014
Ax  Ay
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
10
Examples of Vector Operations
Find the resultant vector which is the sum of A=(2.0i+2.0j) and B =(2.0i-4.0j)
( 4.0) + ( -2.0)
2
2
= 16 + 4.0 = 20 = 4.5(m)
q=
tan
-1
Cy
Cx
=
Find the resultant displacement of three consecutive displacements:
d1=(15i+30j +12k)cm, d2=(23i+14j -5.0k)cm, and d3=(-13i+15j)cm
Magnitude
Tuesday, June 10, 2014
( 25) + (59) + (7.0)
2
2
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
2
= 65(cm)
11
2D Displacement
Displacement:
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
12
2D Average Velocity
Average velocity is the
displacement divided by
the elapsed time.
t - to
Tuesday, June 10, 2014
=
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
13
2D Instantaneous Velocity
The instantaneous velocity indicates how fast the car
moves and the direction of motion at each instant of time.
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
14
2D Average Acceleration
t - to
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
=
15
Displacement, Velocity, and Acceleration in 2-dim
• Displacement:
• Average Velocity:
How is each of
these quantities
defined in 1-D?
• Instantaneous
Velocity:
• Average
Acceleration
• Instantaneous
Acceleration:
Tuesday, June 10, 2014
PHYS 1441-001, Summer 2014
Dr. Jaehoon Yu
16
Kinematic Quantities in 1D and 2D
Quantities
1 Dimension
Displacement
Dx = x f - xi
Average Velocity
Inst. Velocity
Average Acc.
Inst. Acc.
Tuesday, June 10,
2014 is
What
2 Dimension
Dx x f - xi
vx =
=
Dt
t f - ti
Dx
vx º lim
Dt ®0 Dt
Dvx vxf - vxi
ax º
=
Dt
t f - ti
ax º lim
Dt®0
Dvx
Dt
PHYS 1441-001,
Summer
the difference
between
1D 2014
and 2D quantities?
Dr. Jaehoon Yu
17